**Cases where finding the minimum entropy coloring of**

**Cases where finding the minimum entropy coloring of**

**a characteristic graph is a polynomial time problem**

**a characteristic graph is a polynomial time problem**

### The MIT Faculty has made this article openly available.

** Please share **

**Please share**

### how this access benefits you. Your story matters.

**Citation**

### Feizi, Soheil, and Muriel Medard. “Cases Where Finding the

### Minimum Entropy Coloring of a Characteristic Graph Is a Polynomial

### Time Problem.” IEEE, 2010. 116–120. © Copyright 2010 IEEE

**As Published**

### http://dx.doi.org/10.1109/ISIT.2010.5513270

**Publisher**

### Institute of Electrical and Electronics Engineers (IEEE)

**Version**

### Final published version

**Citable link**

### http://hdl.handle.net/1721.1/74017

**Terms of Use**

### Article is made available in accordance with the publisher's

### policy and may be subject to US copyright law. Please refer to the

### publisher's site for terms of use.

## Cases Where Finding the Minimum Entropy

## Coloring of a Characteristic Graph is a Polynomial

## Time Problem

Soheil Feizi, Muriel M´edardRLE at MIT

Emails:*{sfeizi,medard}@mit.edu*

*Abstract—In this paper, we consider the problem of ﬁnding*
the minimum entropy coloring of a characteristic graph under
some conditions which allow it to be in polynomial time. This
problem arises in the functional compression problem where the
computation of a function of sources is desired at the receiver.
The rate region of the functional compression problem has been
considered in some references under some assumptions. Recently,
Feizi et al. computed this rate region for a general one-stage tree
network and its extension to a general tree network. In their
proposed coding scheme, one needs to compute the minimum
entropy coloring (a coloring random variable which minimizes
the entropy) of a characteristic graph. In general, ﬁnding this
coloring is an NP-hard problem (as shown by Cardinal et
al.) . However, in this paper, we show that depending on the
characteristic graph’s structure, there are some interesting cases
where ﬁnding the minimum entropy coloring is not NP-hard,
but tractable and practical. In one of these cases, we show
that, having a non-zero joint probability condition on RVs’
distributions, for any desired function *f, makes characteristic*
graphs to be formed of some non-overlapping fully-connected
maximal independent sets. Therefore, the minimum entropy
coloring can be solved in polynomial time. In another case, we
show that if*f is a quantization function, this problem is also*
tractable.

*Index Terms—Functional compression, graph coloring, graph*
entropy.

I. INTRODUCTION

While data compression considers the compression of sources at transmitters and their reconstruction at receivers, functional compression does not consider the recovery of whole sources, but the computation of a function of sources at the receiver(s).

This problem has been considered in different references
(e.g., [1], [2], [3], [4]). In a recent work in [5], we showed
that, for any one-stage tree network, if source nodes send
-colorings of their characteristic graphs satisfying a condition
*called the Coloring Connectivity Condition (C.C.C.), followed*
by a Slepian-Wolf encoder, it will lead to an achievable coding
scheme. Conversely, any achievable scheme induces colorings
on high probability subgraphs of characteristic graphs
satisfy-ing C.C.C. An extension to a tree network is also considered
in [5].

In the proposed coding scheme in [5], we need to compute the minimum entropy coloring (a coloring random variable This work was supported by AFOSR award number FA9550-09-1-0196.

which minimizes the entropy) of a characteristic graph. In
general, ﬁnding this coloring is an NP-hard problem ([6]).
However, in this paper, we show that depending on the
charac-teristic graph’s structure, there are certain cases where ﬁnding
the minimum entropy coloring is not NP-hard, but tractable
and practical. In one of these cases, we show that, having
a non-zero joint probability condition on RVs’ distributions,
for any desired function *f, makes characteristic graphs to*
be formed of some non-overlapping fully-connected maximal
independent sets. Then, the minimum entropy coloring can be
solved in polynomial time. In another case, we show that if*f*
is a quantization function, this problem is also tractable.

The rest of the paper is organized as follows. Section II presents the minimum entropy coloring problem statement and the necessary technical background. In Section III, main re-sults of this paper are expressed. An application of this paper’s results in the functional compression problem is explained in Section IV. We conclude the paper in Section V.

II. PROBLEMSETUP

In this section, after expressing necessary deﬁnitions, we formulate the minimum entropy coloring problem.

We start with the deﬁnition of a characteristic graph of
a random variable. Consider the network shown in Figure
1 with two sources with RVs *X*1 and *X*2 such that the
computation of a function (*f(X*_{1}*, X*_{2})) is desired at the
receiver. Suppose these sources are drawn from ﬁnite sets

*X*1 *= {x*11*, x*21*, ..., x|X1|*1 *} and X*2 *= {x*12*, x*22*, ..., x|X2|*2 *}. These*

sources have a joint probability distribution *p(x*_{1}*, x _{k}*). We
express

**n-sequences of these RVs as X**

_{1}*= {X*

_{1}

*i}i=l+n−1*and

_{i=l}**X**

_{2}*= {X*

_{2}

*i}i=l+n−1*with the joint probability distribution

_{i=l}* p(x*1

*2*

**, x***). Without loss of generality, we assume l = 1, and*

to simplify notation, *n will be implied by the context if no*
confusion arises. We refer to the*ith*element of**x***j*as*xji*. We
use**x**1* _{j}*,

**x**2

*,... as different*

_{j}*. We shall drop the superscript when no confusion arises. Since the sequence*

**n-sequences of X**_{j}**(x**1

*2*

**, x***) is drawn i.i.d. according to p(x*1

*, x*2), one can write

* p(x*1

*2) =*

**, x***ni=1p(x1i, x2i*).

*Deﬁnition 1. The characteristic graphG _{X1}= (V_{X1}, E_{X1}) of*

*X*1*with respect to* *X*2*,p(x*1*, x*2*), and function f(X*1*, X*2*) is*

*deﬁned as follows:VX1= X*1*and an edge(x*11*, x*21*) ∈ X*12*is in*

*EX1iff there exists ax*12*∈ X*2*such thatp(x*11*, x*12*)p(x*21*, x*12*) >*

*0 and f(x*1

### X

2f (X1,X2)

### X

1Fig. 1. A network with two sources where the computation of a function is desired at the receiver.

In other words, in order to avoid confusion about the
function *f(X*1*, X*2*) at the receiver, if (x*11*, x*21*) ∈ EX1*, then
descriptions of*x*1_{1}and*x*2_{1}must be different.

Next, we deﬁne a coloring function for a characteristic graph.

*Deﬁnition 2. A vertex coloring of a graph is a function*

*cGX1(X*1*) : VX1* *→ N of a graph GX1* *= (VX1, EX1) such*

*that* *(x*1_{1}*, x*2_{1}*) ∈ EX1* *implies* *cGX1(x*11*) = cGX1(x*21*). The*

*entropy of a coloring is the entropy of the induced distribution*
*on colors. Here,* *p(cGX1(xi*1*)) = p(cG−1X1(cGX1(xi*1*))), where*

*c−1*

*GX1(xi*1*) = {x*

*j*

1 *: cGX1(xj*1*) = cGX1(xi*1*)} for all valid j is*

*called a color class. We also call the set of all valid colorings*
*of a graphGX1* *asCGX1.*

Sometimes one needs to consider sequences of a RV with
length*n. In order to deal with these cases, we can extend the*
deﬁnition of a characteristic graph for vectors of a RV.
*Deﬁnition 3. The* *n-th power of a graph GX1* *is a graph*

*Gn*

**X**1*= (VX1n* *, EX1n* *) such that VX1n* *= X*1*nand***(x**11* , x*21

*) ∈ EX1n*

*when there exists at least one* *i such that (x*1* _{1i}, x*2

_{1i}) ∈ EX1.*We denote a valid coloring ofGn*_{X}_{1} *bycGn*

**X1(X**1*).*

In some problems such as the functional compression
prob-lem, we need to ﬁnd a coloring random variable of a
character-istic graph which minimizes the entropy. The problem is how
to compute such a coloring for a given characteristic graph.
In other words, this problem can be expressed as follows.
Given a characteristic graph*G _{X1}* (or, its

*n-th power, Gn*

_{X}_{1}), one can assign different colors to its vertices. Suppose

*C*is the collection of all valid colorings of this graph,

_{G}_{X1}*G*. Among these colorings, one which minimizes the entropy of the coloring RV is called the minimum-entropy coloring, and we refer to it by

_{X1}*cmin*. In other words,

_{G}_{X1}*cmin*

*GX1* = arg min_{c}

*GX1∈CGX1H(cGX1).*

(1)
The problem is how to compute*cmin _{G}_{X1}* given

*G*.

_{X1}III. MINIMUMENTROPYCOLORING

In this section, we consider the problem of ﬁnding the
minimum entropy coloring of a characteristic graph. The
problem is how to compute a coloring of a characteristic
graph which minimizes the entropy. In general, ﬁnding*cmin _{G}*

*X1*

is an NP-hard problem ([6]). However, in this section, we show that depending on the characteristic graph’s structure, there are some interesting cases where ﬁnding the minimum

x11
x12
x13
w1 w2
x11
x12
x13
w1 w2
(a) _{(b)}

Fig. 2. Having non-zero joint probability distribution, a) maximal inde-pendent sets can not overlap with each other (this ﬁgure is to depict the contradiction) b) maximal independent sets should be fully connected to each other. In this ﬁgure, a solid line represents a connection, and a dashed line means no connection exists.

entropy coloring is not NP-hard, but tractable and practical. In
one of these cases, we show that, by having a non-zero joint
probability condition on RVs’ distributions, for any desired
function*f, ﬁnding cmin _{G}*

*X1* can be solved in polynomial time. In

another case, we show that if*f is a quantization function, this*
problem is also tractable. For simplicity, we consider functions
with two input RVs, but one can easily extend all discussions
to functions with more input RVs than two.

*A. Non-Zero Joint Probability Distribution Condition*

Consider the network shown in Figure 1. Source RVs
have a joint probability distribution *p(x*_{1}*, x*_{2}), and the
re-ceiver wishes to compute a deterministic function of sources
(i.e., *f(X*_{1}*, X*_{2})). In Section IV, we will show that in an
achievable coding scheme, one needs to compute minimum
entropy colorings of characteristic graphs. The question is
how source nodes can compute minimum entropy colorings
of their characteristic graphs *G _{X1}* and

*G*(or, similarly the minimum entropy colorings of

_{X2}*Gn*

_{X}_{1}and

*Gn*

_{X}_{2}, for some

*n).*For an arbitrary graph, this problem is NP-hard ([6]). However, in certain cases, depending on the probability distribution or the desired function, the characteristic graph has some special structure which leads to a tractable scheme to ﬁnd the minimum entropy coloring. In this section, we consider the effect of the probability distribution.

*Theorem 4. Suppose for all(x*1*, x*2*) ∈ X*1*× X*2*,p(x*1*, x*2*) >*

*0. Then, maximal independent sets of the characteristic graph*

*GX1* *(and, its* *n-th power Gn***X**1*, for any* *n) are some *

*non-overlapping fully-connected sets. Under this condition, the*
*minimum entropy coloring can be achieved by assigning*
*different colors to its different maximal independent sets.*

*Proof: SupposeΓ(GX1*) is the set of all maximal

indepen-dent sets of *GX1*. Let us proceed by contradiction. Consider
Figure 2-a. Suppose *w*_{1} and*w*_{2} are two different non-empty
maximal independent sets. Without loss of generality, assume

*x*1

1and*x*21are in*w*1, and*x*21and*x*31are in*w*2. In other words,

these sets have a common element*x*2_{1}. Since*w*_{1} and *w*_{2} are
two different maximal independent sets,*x*1_{1} */∈ w*_{2}and*x*3_{1} */∈ w*_{1}.
Since*x*1_{1}and*x*2_{1}are in*w*_{1}, there is no edge between them in

*GX1*. The same argument holds for*x*21and*x*31. But, we have an

edge between*x*1_{1}and*x*3_{1}, because*w*_{1}and*w*_{2}are two different

x21 x22 x23 x11 x12 x13 f(x1,x2) 1 1 1 2 3 3 3 4 X1 X2

Fig. 3. Having non-zero joint probability condition is necessary for Theorem 4. A dark square represents a zero probability point.

maximal independent sets, and at least there should exist such an edge between them. Now, we want to show that it is not possible.

Since there is no edge between*x*1_{1}and*x*2_{1}, for any*x*1_{2}*∈ X*_{2},

*p(x*1

1*, x*12*)p(x*21*, x*12*) > 0, and f(x*11*, x*12*) = f(x*21*, x*12). A similar

argument can be expressed for *x*2_{1} and *x*3_{1}. In other words,
for any *x*1_{2} *∈ X*2, *p(x*2_{1}*, x*1_{2}*)p(x*3_{1}*, x*1_{2}*) > 0, and f(x*2_{1}*, x*1_{2}) =

*f(x*3

1*, x*12*). Thus, for all x*12 *∈ X*2, *p(x*1_{1}*, x*1_{2}*)p(x*3_{1}*, x*1_{2}*) > 0,*
and *f(x*1_{1}*, x*1_{2}*) = f(x*3_{1}*, x*1_{2}*). However, since x*1_{1} and *x*3_{1} are
connected to each other, there should exist a *x*1_{2} *∈ X*_{2} such
that *f(x*1_{1}*, x*1_{2}*) = f(x*3_{1}*, x*1_{2}) which is not possible. So, the
contradiction assumption is not correct and these two maximal
independent sets do not overlap with each other.

We showed that maximal independent sets cannot have
overlaps with each other. Now, we want to show that they
are also fully connected to each other. Again, let us proceed
by contradiction. Consider Figure 2-b. Suppose *w*1 and *w*2

are two different non-overlapping maximal independent sets.
Suppose there exists an element in *w*_{2} (call it *x*3_{1}) which is
connected to one of elements in *w*_{1} (call it *x*1_{1}) and is not
connected to another element of *w*_{1} (call it *x*2_{1}). By using a
similar discussion to the one in the previous paragraph, it is
not possible. Thus,*x*3_{1}should be connected to *x*2_{1}. Therefore,
if for all *(x*_{1}*, x*_{2}*) ∈ X*_{1}*× X*_{2}, *p(x*_{1}*, x*_{2}*) > 0, then maximal*
independent sets of *G _{X}*

_{1}are some separate fully connected sets. In other words, the complement of

*G*

_{X}_{1}is formed by some non-overlapping cliques. Finding the minimum entropy coloring of this graph is trivial and can be achieved by assign-ing different colors to these non-overlappassign-ing fully-connected maximal independent sets.

This argument also holds for any power of*G _{X}*

_{1}. Suppose

**x**1

1, **x**21 and **x**31 are some typical sequences in *X*1*n*. If **x**11 is

not connected to**x**2_{1}and**x**3_{1}, it is not possible to have**x**2_{1}and

**x**3

1connected. Therefore, one can apply a similar argument to

prove the theorem for *Gn*_{X}_{1}, for some *n. This completes the*
proof.

One should notice that the condition*p(x*1*, x*2*) > 0, for all*

*(x*1*, x*2*) ∈ X*1*× X*2, is a necessary condition for Theorem 4.

In order to illustrate this, consider Figure 3. In this example,

*x*1

1,*x*21and*x*31are in*X*1, and*x*12,*x*22and*x*32are in*X*2. Suppose

*p(x*2

1*, x*22*) = 0. By considering the value of function f at these*

points depicted in the ﬁgure, one can see that, in*GX*1,*x*21 is

not connected to*x*1_{1}and*x*3_{1}. However,*x*1_{1}and*x*3_{1}are connected
to each other. Thus, Theorem 4 does not hold here.

f(x1,x2) (a) X1 1 2 3 4 5 X2 f(x1,x2) (b) 1 X2 1 4 4 2 2 5 5 3 X21 X22 X23 X11 X12 X13 X1 Function Region X12×X23

Fig. 4. a) A quantization function. Function values are depicted in the ﬁgure on each rectangle. b) By extending sides of rectangles, the plane is covered by some function regions.

It is also worthwhile to notice that the condition used in
Theorem 4 only restricts the probability distribution and it
does not depend on the function *f. Thus, for any function*

*f at the receiver, if we have a non-zero joint probability*

distribution of source RVs (for example, when source RVs are independent), ﬁnding the minimum-entropy coloring and, therefore, the proposed functional compression scheme of Section IV and reference [5], is easy and tractable.

*B. Quantization Functions*

In Section III-A, we introduced a condition on the joint probability distribution of RVs which leads to a speciﬁc structure of the characteristic graph such that ﬁnding the minimum entropy coloring is not NP-hard. In this section, we consider some special functions which lead to some graph structures so that one can easily ﬁnd the minimum entropy coloring.

An interesting function is a quantization function. A nat-ural quantization function is a function which separates the

*X*1*− X*2plane into some rectangles such that each rectangle

corresponds to a different value of that function. Sides of these rectangles are parallel to the plane axes. Figure 4-a depicts such a quantization function.

Given a quantization function, one can extend different sides
of each rectangle in the*X*_{1}*− X*_{2}plane. This may make some
*new rectangles. We call each of them a function region. Each*
function region can be determined by two subsets of*X*_{1}and

*X*2. For example, in Figure 4-b, one of the function regions is

distinguished by the shaded area.

*Deﬁnition 5. Consider two function regions* *X*_{1}1*× X*_{2}1 *and*
*X*2

1 *× X*22*. If for anyx*11 *∈ X*11 *andx*21 *∈ X*12*, there existx*2

*such thatp(x*1_{1}*, x*_{2}*)p(x*2_{1}*, x*_{2}*) > 0 and f(x*1_{1}*, x*_{2}*) = f(x*2_{1}*, x*_{2}*),*
*we say these two function regions are pairwiseX*_{1}*-proper.*

*Theorem 6. Consider a quantization function* *f such that*

*its function regions are pairwiseX*_{1}*-proper. Then,G _{X}*

_{1}

*(and*

*Gn*

**X**1*, for any* *n) is formed of some non-overlapping *

*fully-connected maximal independent sets, and its minimum entropy*
*coloring can be achieved by assigning different colors to*
*different maximal independent sets.*

*Proof: We ﬁrst prove it forG _{X}*

_{1}. Suppose

*X*

_{1}1

*× X*

_{2}1, and

*X*2

function *f, where X*_{1}1 *= X*_{1}2. We show that *X*_{1}1 and *X*_{1}2
are two non-overlapping fully-connected maximal independent
sets. By deﬁnition,*X*_{1}1and*X*_{1}2are two non-equal partition sets
of*X*_{1}. Thus, they do not have any element in common.

Now, we want to show that vertices of each of these partition
sets are not connected to each other. Without loss of generality,
we show it for *X*_{1}1. If this partition set of *X*_{1} has only one
element, this is a trivial case. So, suppose*x*1_{1}and*x*2_{1}are two
elements in *X*_{1}1. By deﬁnition of function regions, one can
see that, for any *x*2*∈ X*2 such that*p(x*_{1}1*, x*2*)p(x*21*, x*2*) > 0,*

then*f(x*1_{1}*, x*_{2}*) = f(x*2_{1}*, x*_{2}). Thus, these two vertices are not
connected to each other. Now, suppose*x*3_{1}is an element in*X*_{1}2.
Since these function regions are*X*1-proper, there should exist
at least one*x*_{2} *∈ X*_{2}, such that *p(x*_{1}1*, x*_{2}*)p(x*3_{1}*, x*_{2}*) > 0, and*

*f(x*1

1*, x*2*) = f(x*31*, x*2*). Thus, x*11and*x*31are connected to each

other. Therefore, *X*_{1}1 and*X*_{1}2 are two non-overlapping
fully-connected maximal independent sets. One can easily apply this
argument to other partition sets. Thus, the minimum entropy
coloring can be achieved by assigning different colors to
different maximal independent sets (partition sets). The proof
for*Gn*_{X}_{1}, for any*n, is similar to the one mentioned in Theorem*
4. This completes the proof.

It is worthwhile to mention that without *X*_{1}-proper
con-dition of Theorem 6, assigning different colors to different
partitions still leads to an achievable coloring scheme.
How-ever, it is not necessarily the minimum entropy coloring. In
other words, without this condition, maximal independent sets
may overlap.

*Corollary 7. If a functionf is strictly increasing (or, *

*decreas-ing) with respect toX*_{1}*, andp(x*_{1}*, x*_{2}*) = 0, for all x*_{1}*∈ X*_{1}
*andx*_{2} *∈ X*_{2}*, then,* *G _{X}*

_{1}

*(and,*

*Gn*

_{X}_{1}

*for any*

*n) would be a*

*complete graph.*

Under conditions of Corollary 7, functional compression
does not give us any gain, because, in a complete graph, one
should assign different colors to different vertices. Traditional
compression in which *f is the identity function is a special*
case of Corollary 7.

IV. APPLICATIONS IN THEFUNCTIONALCOMPRESSION

PROBLEM

As we expressed in Section II, results of this paper can be used in the functional compression problem. In this section, after giving the functional compression problem statement, we express some of previous results. Then, we explain how this paper’s results can be applied in this problem.

*A. Functional Compression Problem Setup*

Consider the network shown in Figure 1. In this network, we
have two source nodes with input processes*{X _{j}i}∞_{i=1}*for

*j =*

*1, 2. The receiver wishes to compute a deterministic function*

*f : X*1*×X*2*→ Z, or f : X*1*n×X*2*n→ Zn*, its vector extension.

Source node *j encodes its message at a rate R _{X}_{j}*. In
other words, encoder

*en*maps

_{X}_{j}*X*

_{j}n*→ {1, ..., 2nRXj*The receiver has a decoder

_{}.}*r, which maps {1, ..., 2nRX1} ×*

*{1, ..., 2nRX2} → Zn*. We sometimes refer to this
encod-ing/decoding scheme as an*n-distributed functional code.*

For any encoding/decoding scheme, the probability
of error is deﬁned as *P _{e}n* =

*P r*

**(x**1

*2) :*

**, x*** f(x*1

*2*

**, x***) = r(enX*1

**(x**1

*), enX*2

**(x**2))

. A rate pair,

*R = (RX*1*, RX*2) is achievable iff there exist encoders

in source nodes operating in these rates, and a decoder*r at*
the receiver such that *P _{e}n*

*→ 0 as n → ∞. The achievable*rate region is the set closure of the set of all achievable rates.

*B. Prior Results in the Functional Compression Problem*

In order to express some previous results, we need to deﬁne
an*-coloring of a characteristic graph.*

*Deﬁnition 8. Given a non-empty set* *A ⊂ X*_{1}*× X*_{2}*, *
*de-ﬁne* *ˆp(x*_{1}*, x*_{2}*) = p(x*_{1}*, x*_{2}*)/p(A) when (x*_{1}*, x*_{2}*) ∈ A, and*

*ˆp(x*1*, x*2*) = 0 otherwise. ˆp is the distribution over (x*1*, x*2)

*conditioned on(x*_{1}*, x*_{2}*) ∈ A. Denote the characteristic graph*
*ofX*_{1}*with respect toX*_{2}*,ˆp(x*_{1}*, x*_{2}*), and f(X*_{1}*, X*_{2}*) as ˆG _{X}*

_{1}=

( ˆ*VX*1*, ˆEX*1*) and the characteristic graph of X*2 *with respect*

*toX*_{1}*,* *ˆp(x*_{1}*, x*_{2}*), and f(X*_{1}*, X*_{2}*) as ˆG _{X}*

_{2}= ( ˆ

*V*

_{X}_{2}

*, ˆE*

_{X}_{2}

*). Note*

*that ˆE*

_{X}_{1}

*⊆ E*

_{X}_{1}

*and ˆE*

_{X}_{2}

*⊆ E*

_{X}_{2}

*. Finally, we say that*

*cG*1

_{X1}(X*) and cG*2

_{X2}(X*) are -colorings of GX*1

*andGX*2

*if*

*they are valid colorings of ˆG _{X}*

_{1}

*and ˆG*

_{X}_{2}

*deﬁned with respect*

*to some setA for which p(A) ≥ 1 − .*

Among *-colorings of G _{X}*

_{1}, the one which minimizes the entropy is called the minimum entropy

*-coloring of G*

_{X}_{1}. Sometimes, we refer to it as the minimum entropy coloring when no confusion arises.

Consider the network shown in Figure 1. The rate region of this network has been considered in different references under some assumptions. Reference [3] considered the case where

*X*2is available at the receiver as the side information, while

reference [4] computed a rate region under some conditions on source distributions called the zigzag condition. The case of having more than one desired function at the receiver with the side information was considered in [7].

However, the rate region for a more general case has
been considered in [5]. Reference [5] shows that if source
nodes send *-colorings of their characteristic graphs *
*satisfy-ing a condition called the Colorsatisfy-ing Connectivity Condition*
(C.C.C.), followed by a Slepian-Wolf encoder, it will lead
to an achievable coding scheme. Conversely, any achievable
scheme induces colorings on high probability subgraphs of
characteristic graphs satisfying C.C.C.

In the following, we brieﬂy explain C.C.C. for the case of having two source nodes. An extension to the case of having more source nodes than one is presented in [5].

*Deﬁnition 9. A path with lengthm between two points Z*_{1}=
*(x*1

1*, x*12*) and Zm* *= (x*21*, x*22*) is determined by m points Zi,*

*1 ≤ i ≤ m such that,*

*i)P (Z _{i}) > 0, for all 1 ≤ i ≤ m.*

*ii)Z _{i}andZ_{i+1}only differ in one of their coordinates.*

Deﬁnition 9 can be expressed for two *n-length vectors*
**(x**1

1* , x*12

**) and (x**21

*22).*

**, x***Deﬁnition 10. A joint-coloring family* *JC* *for random *
*vari-ables* *X*_{1}*and* *X*_{2} *with characteristic graphsG _{X}*

_{1}

*andG*

_{X}_{2}

*,*

*and any valid coloringsc*

_{G}_{X1}andc_{G}_{X2}, respectively is deﬁnedx11 x12 x21 x22 f(x1,x2) 0 0 0 x11 x12 x21 x22 f(x1,x2) 1 0 (a) (b)

Fig. 5. Two examples of a joint coloring class: a) satisfying C.C.C. b) not satisfying C.C.C. Dark squares indicate points with zero probability. Function values are depicted in the picture.

*as* *J _{C}*

*= {j*1

_{c}*, ..., jcnjc} where jci*=

*(xi1*1

*, xj1*2

*), (xi2*1

*, xj2*2) :

*cGX1(xi1*1

*) = cGX1(xi2*1

*), cGX2(xj1*2

*) = cGX2(xj2*2)

*for any*

*validi*

_{1}

*,i*

_{2}

*,j*

_{1}

*andj*

_{2}

*, wheren*

_{j}_{c}= |c_{G}_{X1}| × |c_{G}_{X2}|. We call*eachj*

_{c}ias a joint coloring class.Deﬁnition 10 can be expressed for RVs **X**_{1} and **X**_{2} with
characteristic graphs*Gn*_{X}_{1}and*Gn*_{X}_{2}, and any valid*-colorings*

*cGn*

**X1** and*cGn***X2**, respectively.

*Deﬁnition 11. Consider RVsX*_{1} *andX*_{2}*with characteristic*
*graphs* *G _{X1}*

*and*

*G*

_{X2}, and any valid colorings*c*

_{G}_{X1}*and*

*cGX2. We say these colorings satisfy the Coloring Connectivity*

*Condition (C.C.C.) when, between any two points inj _{c}i∈ J_{C},*

*there exists a path that lies inj*

_{c}i, or functionf has the same*value in disconnected parts ofj*

_{c}i.C.C.C. can be expressed for RVs**X**_{1} and**X**_{2}with
charac-teristic graphs*Gn*_{X}_{1} and*Gn*_{X}_{2}, and any valid*-colorings c _{G}n*

**X1**
and*c _{G}n*

**X2**, respectively.

In order to illustrate this condition, we borrow an example from [5].

*Example 12. For example, suppose we have two random*

*variables* *X*_{1} *and* *X*_{2} *with characteristic graphs* *G _{X1}*

*and*

*GX2. Let us assumecGX1*

*andcGX2*

*are two valid colorings*

*ofG _{X1}andG_{X2}, respectively. Assumec_{G}_{X1}(x*1

_{1}

*) = c*2

_{G}_{X1}(x_{1})

*and*

*c*1

_{G}_{X2}(x_{2}

*) = c*2

_{G}_{X2}(x_{2}

*). Suppose j*1

_{c}*represents this joint*

*coloring class. In other words,*

*j*1

_{c}*= {(xi*

_{1}

*, xj*

_{2}

*)}, for all*

*1 ≤ i, j ≤ 2 when p(xi*

1*, xj*2*) > 0. Figure 5 considers two*

*different cases. The ﬁrst case is whenp(x*1_{1}*, x*2_{2}*) = 0, and other*
*points have a non-zero probability. It is illustrated in Figure *
*5-a. One can see that there exists a path between any two points*
*in this joint coloring class. So, this joint coloring class satisﬁes*
*C.C.C. If other joint coloring classes ofc _{G}_{X1}*

*andc*

_{G}_{X2}*satisfy*

*C.C.C., we sayc*

_{G}_{X1}*andc*

_{G}_{X2}*satisfy C.C.C. Now, consider*

*the second case depicted in Figure 5-b. In this case, we have*

*p(x*1

1*, x*22*) = 0, p(x*21*, x*12*) = 0, and other points have a *

*non-zero probability. One can see that there is no path between*

*(x*1

1*, x*12*) and (x*21*, x*22*) in jc*1*. So, though these two points belong*

*to a same joint coloring class, their corresponding function*
*values can be different from each other. For this example,*
*j*1

*c* *does not satisfy C.C.C. Therefore,cGX1* *andcGX2* *do not*

*satisfy C.C.C.*

In Section III, we did not consider C.C.C. in ﬁnding the minimum entropy coloring. For the non-zero joint probability

case, under the condition mentioned in Theorem 4, any valid colorings (and therefore, minimum entropy colorings) of char-acteristic graphs satisfy C.C.C mentioned in Deﬁnition 11. It is because we have at least one path between any two points. However, for a quantization function, after achieving minimum entropy colorings, C.C.C. should be checked for colorings. In other words, despite the non-zero joint probability condition where C.C.C. always holds, for the quantization function case, one should check C.C.C. separately.

V. CONCLUSION

In this paper we considered the problem of ﬁnding the
minimum entropy coloring of a characteristic graph under
some conditions which make it to be done in polynomial time.
This problem has been raised in the functional compression
problem where the computation of a function of sources is
desired at the receiver. Recently, [5] computed the rate region
of the functional compression problem for a general one stage
tree network and its extension to a general tree network. In
the proposed coding schemes of [5] and some other references
such as [4] and [7], one needs to compute the minimum
entropy coloring of a characteristic graph. In general, ﬁnding
this coloring is an NP-hard problem ([6]). However, in this
paper, we showed that, depending on the characteristic graph’s
structure, there are some interesting cases where ﬁnding the
minimum entropy coloring is not NP-hard, but tractable and
practical. In one of these cases, we show that, by having a
non-zero joint probability condition on RVs’ distributions, for
any desired function*f, ﬁnding the minimum entropy coloring*
can be solved in polynomial time. In another case, we show
that, if*f is a type of a quantization function, this problem is*
also tractable.

In this paper, we considered two cases such that ﬁnding the minimum entropy coloring of a characteristic graph can be done in polynomial time. Depending on different applications, one can investigate other conditions on either probability distributions or the desired function to have a special structure of the characteristic graph which may lead to polynomial time algorithms to ﬁnd the minimum entropy coloring.

REFERENCES

[1] J. K¨orner, “Coding of an information source having ambiguous alphabet and the entropy of graphs,” 6th Prague Conference on Information Theory, 1973, pp. 411–425.

[2] H. S. Witsenhausen, “The zero-error side information problem and
*chromatic numbers,” IEEE Trans. Inf. Theory, vol. 22, no. 5, pp. 592–593,*
Sep. 1976.

*[3] A. Orlitsky and J. R. Roche, “Coding for computing,” IEEE Trans. Inf.*

*Theory, vol. 47, no. 3, pp. 903–917, Mar. 2001.*

[4] V. Doshi, D. Shah, M. M´edard, and S. Jaggi, “Distributed functional
*com-pression through graph coloring,” in 2007 Data Comcom-pression Conference,*
Snowbird, UT, Mar. 2007, pp. 93–102.

[5] S. Feizi and M. M´edard, “When do only sources need to compute? on functional compression in tree networks,” invited paper 2009 Annual Allerton Conference on Communication, Control, and Computing, Sep. 2009.

[6] J. Cardinal, S. Fiorini, and G. Joret, “Tight results on minimum entropy set cover,” vol. 51, no. 1, pp. 49–60, May 2008.

[7] S. Feizi and M. M´edard, “Multifunctional compression with side
*infor-mation,” in 2009 Globecom Conference on Communications, HI, US,*
Nov.-Dec. 2009.